everywhere in nature runaway sexual selection involves females selecting males for a disproportionate value of some characteristic, never the other way around.
Although I have said “chooser” and “chosen” wherever possible, the terms map onto female and male in most species. Male choice does occur, however, in monogamous (or quasi-monogamous) species like sea horses and humans, where both mates have a similar interest in their brood and are willing to spend time assessing, and being assessed by, potential mates. A very few species thoroughly reverse the usual pattern, making females compete for access to males. Mormon cricket females in food-poor areas actually fight to obtain males’ packages of edible sperm, while the coy males refuse about half the copulations they are offered. Among jacanas, a lily-trotting waterbird, males incubate the eggs while females fight with one another and guard their current males. Moorhen females are larger and brighter than males and more likely to fight and court. Small moorhen males are actually in better condition than big ones, and spend more time incubating. Females compete most for small fat males!
It’s almost that absolute, but not quite. Regarding role reversal in sexual selection, Alison Jolly writes in Lucy’s Legacy: Sex and Intelligence in Human Evolution, page 88-89: